Superfast diffusion exists in various complex anisotropic systems. Its mean square displacement is an exponential function of time proved by several theoretical and experimental investigations. Previous studies have studied the superfast diffusion based on the time-space scaling local structural derivatives without considering the memory of dynamic behavior. This paper proposes a nonlocal time structural derivative model based on the Caputo fractional derivative to describe superfast diffusion in which the structural function is a power law function of time. The obtained concentration of the diffusive particles, i.e. the solution of the structural derivative model is a double-sided exponential distribution. The derived mean square displacement is a Mittag–Leffler function of time, which generalizes the exponential case. To verify the feasibility of the model, the charge and energy transfer at nanoscale interfaces in solar cells and the dynamics of the dripplons between two graphene sheets are employed. Compared with the existing models, the fitting results indicate that the proposed model is more accurate with higher credibility. The properties of the nonlocal structural derivative model with different structural functions are also discussed.
Goal-oriented adaptive modeling of random heterogeneous media and model-based multilevel Monte Carlo methods
